Backpropagation of neural nets with shared weight
If we write $ H^2 = W^{h}H1 + W^{x}X^{2} $ then it will be better to understand the backward propagation step.

Now,

$\frac{\partial}{\partial W^{x}} W^{h}W^{x}X^{1}$ can be written as:
$\frac{\partial H^2}{\partial H^1}\frac{\partial H^1}{\partial W^{x}} $

$\frac{\partial H^2}{\partial H^1} =  (W^h)^T$ and 
$\frac{\partial H^1}{\partial W^{x}} = (X^{1})^T $

Therefore,

$\frac{\partial}{\partial W^{x}} W^{h}W^{x}X^{1} = (W^h)^T(X^{1})^T  $

I hope it has solved your problem.